According to the Miessler Tarr and many molecular orbital diagrams for octahedral complexes on the internet will σ-donors/acceptors only influence the $e_g$-orbital, while π-donors/acceptors influence the $t_2g$-orbitals. This can also be shown using the corresponding character table, but it's been a while since I used these, so I will just believe its correct. Still, I was wondering, is there no case of it ever mixing, for example, that a π-donor/acceptor does not coordinate with both lobes but in a σ-fashion? I thought about this case, and there is also an example mentioned in the book, where they show how a σ-donor and π-acceptor-ligand (I suppose they mean one with both abilities) would look like. Here they draw the lines from the π-orbitals or rather π*-orbitals in this case to the $t_2g$-orbital and the σ-orbitals are connected to the $e_g$-orbital.
One example where I know this behavior, just vice versa would be chloride as a ligand, and I have also seen schemes, where it was drawn to coordinate in a π-fashion to the metal's d-orbitals. But in this case, you have both properties. What about a hydride for example, as pure σ-ligand? Would it only influence the $e_g$-orbital or is there a way it could also somehow weakly influence the $t_2g$-orbital?